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On distinguished square-integrable representations for
Galois pairs and a conjecture of Prasad
Raphaël Beuzart-Plessis
To cite this version:
Raphaël Beuzart-Plessis. On distinguished square-integrable representations for Galois pairs and a conjecture of Prasad. Inventiones Mathematicae, Springer Verlag, 2018, 214 (1), pp.437-521. �10.1007/s00222-018-0807-z�. �hal-01941337�
On distinguished square-integrable representations for
Galois pairs and a conjecture of Prasad
Raphaël Beuzart-Plessis
∗November 30, 2018
Abstract
We prove an integral formula computing multiplicities of square-integrable repre-sentations relative to Galois pairs over p-adic fields and we apply this formula to verify two consequences of a conjecture of Dipendra Prasad. One concerns the exact compu-tation of the multiplicity of the Steinberg represencompu-tation and the other the invariance of multiplicities by transfer among inner forms.
Contents
Introduction 1
1 Preliminaries 8
2 Definition of a distribution for all symmetric pairs 30
3 The spectral side 40
4 The geometric side 45
5 Applications to a conjecture of Prasad 58
Acknowledgment 73
Introduction
Let F be a p-adic field (that is a finite extension of Qp for a certain prime number p)
and H be a connected reductive group over F . Let E{F be a quadratic extension and set
∗Université d’Aix-Marseille, I2M-CNRS(UMR 7373), Campus de Luminy, 13288 Marseille Cédex 9, France
G :“ RE{FHE where RE{F denotes Weil’s restriction of scalars (so that GpF q “ HpEq). To
every complex smooth irreducible representation π of GpF q and every continuous character χ of HpF q we associate a multiplicity mpπ, χq (which is always finite by [18] Theorem 4.5) defined by
mpπ, χq :“ dim HomHpπ, χq
where HomHpπ, χq stands for the space of pHpF q, χq-equivariant linear forms on (the space
of) π. Recently, Dipendra Prasad [43] has proposed very general conjectures describing this multiplicity in terms of the Langlands parameterization of π, at least for representations belonging to the so-called ‘generic’ L-packets. These predictions, which generalize earlier conjectures of Jacquet ([27], [28]), are part of a larger stream that has come to be called the ‘local relative Langlands program’ and whose main aim is roughly to describe the ‘spectrum’ of general homogeneous spherical varieties X “ HzG over local fields in terms of Langlands dual picture and correspondence. In the paper [44], and under the assumption that G is split, Sakellaridis and Venkatesh set up a very general framework to deal with these questions by introducing a certain complex reductive group ˇGX associated to the variety X, which
generalizes Langlands construction of a dual group, together with a morphism ˇGX Ñ ˇG
(actually, in the most general case, this should also include an extra SL2 factor) which,
according to them, should govern a great part of the spectral decomposition of L2pHzGq
(see [44] Conjecture 16.2.2). In a similar way, in the case where G “ RE{FHE as above
(note that such a group is never split) Prasad introduces a certain L-group LHop (further
explanations on this notation are given below) and a morphism LHop Ñ LG which should
govern, on the dual side, the behavior of the multiplicities mpπ, χq for a very particular quadratic character χ (denoted by ωH,E below) that has also been defined by Prasad. The
main goal of this paper is to present some coarse results supporting Prasad’s very precise conjectures in the particular case of stable (essentially) square-integrable representations. In the rest of this introduction we will recall the part of Prasad’s conjecture that we are interested in as well as the two consequences of it that we have been able to verify. We will also say some words on the proofs which are based on a certain simple local trace formula adapted to the situation and which takes its roots in Arthur’s local trace formula ([4]) as well as in Waldspurger’s work on the Gross-Prasad conjecture for orthogonal groups ([46], [47]).
Prasad associates a number of invariants to the situation at hand. First, he constructs a certain quadratic character ωH,E : HpF q Ñ t˘1u as well as a certain quasi-split group Hop
over F which is an E{F form of the quasi-split inner form of H. We refer the reader to [43]§7-8 for precise constructions of those and content ourself to give three examples here:
• If H “ GLn, then Hop “ Upnqqs (quasi-split form) and ωH,E “ pηE{F ˝ detqn`1 where
ηE{F is the quadratic character associated to E{F ;
• If H “ Upnq (a unitary group of rank n), then Hop “ GL
• If H “ SOp2n ` 1q (any odd special orthogonal group), then Hop “ SOp2n ` 1q qs
(the quasi-split inner form) and ωH,E “ ηE{F ˝ Nspin where Nspin : SOp2n ` 1qpF q Ñ
Fˆ{pFˆq2 denotes the spin norm.
To continue we need to restrict slightly the generality by only considering characters χ that are of ‘Galois type’ i.e. which are in the image of a map constructed by Langlands
H1pWF, Zp ˇHqq Ñ HomcontpHpF q, Cˆq
This map is always injective (because F is p-adic) but not always surjective (although it is most of the time, e.g. if H is quasi-split). We refer the reader to [37] for further discussion on these matters. The character ωH,E is always of Galois type and, to every character χ of
Galois type of HpF q, Prasad associates a certain ‘Langlands dual group’ Hop
χ which sits in a
short exact sequence
1Ñ ˇHop Ñ Hop
χ Ñ WF Ñ 1
together with a group embedding ι : Hop
χ ãÑ LG (where LG denotes the L-group of G)
compatible with the projections to WF and algebraic when restricted to ˇHop. In the
partic-ular case where χ “ ωH,E, we have Hopχ “ LHop and ι is the homomorphism of quadratic
base-change.
Remark 1 Although the short exact sequence above always splits, there does not necessarily exist a splitting preserving a pinning of ˇHop and hence Hop
χ is not always an L-group in the
usual sense.
Let W DF :“ WF ˆ SL2pCq be the Weil-Deligne group of F . An ‘L-parameter’ taking
values in Hop
χ is defined as usual: a continuous Frobenius semi-simple morphism W DF Ñ Hopχ
which commutes with the projections to WF and is algebraic when restricted to SL2pCq. We
are now ready to state (a slight generalization of) the stable version of Prasad’s conjecture for square-integrable representations:
Conjecture 1 Let φ : W DF Ñ LG be a discrete L-parameter, ΠGpφq Ď IrrpGq the
corre-sponding L-packet and Πφ “
ÿ
πPΠGpφq
dpπqπ the stable representation associated to φ. Then, we have mpΠφ, χq “ |ker1pF ; H, Gq|´1 ÿ ψ |Zpφq| |Zpψq| where
• The sum is over the set of ‘L-parameters’ ψ : W DF Ñ Hopχ (taken up to ˇHop-conj)
making the following diagram commute up to ˇG-conj, i.e. there exists g P ˇG such that ι˝ ψ “ gφg´1,
Hop χ _ ι W DF ψ ①;; ① ① ① ① ① ① ① φ //LG
• ker1pF ; H, Gq :“ Ker pH1pF, Hq Ñ H1pF, Gqq (corresponds to certain twists of the
pa-rameter ψ that become trivial in LG);
• Zpφq :“ CentGˇpφq{Zp ˇGqWF and Zpψq :“ CentHˇoppψq{Zp ˇHopqWF
As we said, this is only part of Prasad’s general conjectures which aim to describe (almost) all the multiplicities mpπ, χq explicitly. This version of the conjecture (and far more) is known in few particular cases: for H “ GLpnq by Kable and Anandavardhanan-Rajan ([29], [1]), for H “ Upnq by Feigon-Lapid-Offen ([21]) and for H “ GSpp4q by Hengfei Lu ([39]). The following theorems are both formal consequences of Conjecture 1 and are the main results of this paper (see Theorem 5.3.1 and Theorem 5.7.1):
Theorem 1 Let H, H1 be inner forms over F , G :“ R
E{FH, G1 :“ RE{FH1 and χ, χ1
characters of Galois type of HpF q and H1pF q corresponding to each other (i.e. coming from
the same element in H1pW
F, Zp ˇHqq “ H1pWF, Zp ˇH1qq). Let Π, Π1 be (essentially)
square-integrable representations of GpF q and G1pF q respectively which are stable (but not necessarily
irreducible) and transfer of each other (i.e. ΘΠpxq “ ΘΠ1pyq for all stably conjugate regular
elements xP GregpF q and y P G1regpF q where ΘΠ, ΘΠ1 denote the Harish-Chandra characters
of Π and Π1 respectively). Then, we have
mpΠ, χq “ mpΠ1, χ1q
Theorem 2 For π“ StpGq the generalized Steinberg representation of GpF q and χ a char-acter of Galois type we have
mpStpGq, χq “ "
1 if χ“ ωH,E
0 otherwise
Theorem 2 also confirms an older conjecture of Prasad ([42], Conjecture 3) which was al-ready known for split groups and tamely ramified extensions by work of Broussous-Courtès and Courtès ([12], [15], [16]) and for inner forms of GLn by work of Matringe ([40]). The
proof of Broussous and Courtès is mainly based on a careful study of the geometry of the building whereas Matringe’s work uses some Mackey machinery. Our approach is completely orthogonal to theirs and is based on a certain integral formula computing the multiplicity mpπ, χq in terms of the Harish-Chandra character of π. This formula is reminiscent and inspired by a similar result of Waldspurger in the context of the so-called Gross-Prasad con-jecture ([46], [47]). It can also be seen as a ‘twisted’ version (‘twisted’ with respect to the non-split extension E{F ) of the orthogonality relations between characters of discrete series due Harish-Chandra ([14], Theorem 3). It can be stated as follows (see Theorem 5.1.1):
Theorem 3 Let π be a square-integrable representation of GpF q and χ be a continuous character of HpF q. Assume that χ and the central character of π coincide on AHpF q (the
maximal split central torus in HpF q). Then, we have
mpπ, χq “ ż
ΓellpHq
DHpxqΘπpxqχpxq´1dx
where Θπ denotes the Harish-Chandra character of π (a locally constant function on GregpF q),
DH is the usual Weyl discriminant and Γ
ellpHq stands for the set of regular elliptic conjugacy
classes in HpF q :“ HpF q{AHpF q equipped with a suitable measure dx.
Theorem 1 is an easy consequence of this formula and Theorem 2 also follows from it with some extra work. Let us give an outline of the proof of Theorem 2 assuming Theorem 3. For notational simplicity we will assume that H is semi-simple. We have the following explicit formula for the character of the Steinberg representation (see §5.5 for a reminder)
DGpxq1{2ΘStpGqpxq “ ÿ
P0ĎP “M U
p´1qaP´aP0 ÿ
tyPM pF q; y„conjxu{M ´conj
DMpyq1{2δPpyq1{2
where P0 is a minimal parabolic subgroup of G and we refer the reader to the core of the
paper for other unexplained notations which are however pretty standard. Plugging this explicit formula in Theorem 3 and rearranging somewhat the terms we get
mpStpGq, χq “ ÿ pM,P q{conj p´1qaP´aP0 ż ΓellpMq DMpxqχpxq´1dx
where the sum runs over the HpF q-conjugacy classes of pairs pM, P q with
• M an elliptic twisted Levi subgroup of H by which we mean an algebraic subgroup of H with trivial split center such that RE{FME is a Levi subgroup of G;
• P a parabolic subgroup of G with Levi component RE{FME.
Using a particular case of Harish-Chandra orthogonality relations between characters of discrete series ([14] Theorem 3), we can show that (see §5.6)
ż
ΓellpMq
DMpxqχpxq´1dx“ pχ|M, 1q
where χ|M denotes the restriction of χ to MpF q, 1 the trivial character of MpF q and p., .q
denotes the natural scalar product on the space of virtual characters of MpF q. Then, in the above expression for mpStpGq, χq, we can group together pairs pM, P q according to their stable conjugacy classes ending up with an equality
mpStpGq, χq “ ÿ
pM,P q{stab
p´1qaP´aP0|ker1pF ; M, Hq|pχ
where ker1pF ; M, Hq :“ Ker pH1pF, Mq Ñ H1pF, Hqq, a set which naturally parametrizes
conjugacy classes inside the stable conjugacy class of pM, P q. Set Hab for the quotient of
HpF q by the common kernel of all the characters of Galois type (in case H is quasi-split it is just the abelianization of HpF q) and let Mab denote, for all elliptic twisted Levi M, the
image of MpF q in Hab. Then, using Frobenius reciprocity, the last identity above can be
rewritten as the equality between mpStpGq, χq and ¨ ˝χ, ÿ pM,P q{stab p´1qaP´a ker1pF ; M, Hq Ind Hab Mabp1q ˛ ‚
and thus Theorem 2 is now equivalent to the following identity in the Grothendieck group of Hab: ÿ pM,P q{stab p´1qaP´a ker1pF ; M, Hq Ind Hab Mabp1q “ ωH,E (0.0.1)
The proof of this identity in general is rather long and technical (see Proposition 5.4.1), so we content ourself (again) with giving two examples here:
• If H “ GLn, we have Hab “ Fˆ and ωH,E “ ηE{Fn`1. If n is odd, M “ H is the only
elliptic twisted Levi and then 0.0.1 reduces to 1 “ 1. On the other hand, if n is even there are two (stable) conjugacy classes of pairs pM, P q:
M0 “ P0 “ H and M1 “ GLn{2pEq Ď P1 “
ˆ
GLn{2pEq ˚
GLn{2pEq
˙
we have M0,ab“ Hab “ Fˆ Ą M1,ab “ NpEˆq and 0.0.1 reduces to the identity
IndFNpEˆ ˆq1´ 1 “ ηE{F
• For H “ Upnq (a unitary group of rank n) we have Hab “ Ker NE{F and ωH,E “ 1.
In this case, stable conjugacy classes of pairs pM, P q are parametrized by (ordered) partitions pn1, . . . , nkq of n as follows: pn1, . . . , nkq ÞÑ M “ Upn1q ˆ . . . ˆ Upnkq Ď P “ ¨ ˚ ˝ GLn1pEq ˚ ˚ . .. ˚ GLnkpEq ˛ ‹ ‚
Moreover, |ker1pF ; M, Hq| “ 2k´1 and M
ab “ Hab for all M as above. Thus, in this
ÿ
pn1,...,nkq
n1`...`nk“n
p´1qn´k2k´1 “ 1
As we said, for its part, Theorem 3 is a consequence of a certain simple local trace formula adapted to the situation and to the proof of which most of the paper is devoted. Let us state briefly the content of this formula by assuming again, for simplicity, that the group H is semi-simple. Starting with a function f P C8
c pGpF qq, we consider the following expression
in two variables
Kfχpx, yq :“ ż
HpF q
fpx´1hyqχphq´1dh, x, y P GpF q
This function is precisely the kernel of the operator on L2pHpF qzGpF q, χq given by
convo-lution by f. Formally, the trace of such an operator should be given by the integral of this kernel over the diagonal that is
Jχpf q :“ ż
HpF qzGpF q
Kfχpx, xqdx
Unfortunately, in general the convolution operator given by f isn’t of trace-class and the above expression diverges. Nevertheless, we can still restrict our attention to some ‘good’ space of test functions for which the above integral is absolutely convergent. Recall, following Waldspurger [46], that the function f is said to be strongly cuspidal if for all proper parabolic subgroups P “ MU Ĺ G we have
ż
UpF q
fpxuqdu “ 0
for all x P MpF q. We also say, following Harish-Chandra [23], that f is a cusp form if the above kind of integrals vanish for all x P GpF q (thus, and contrary to what we might guess, being a cusp form is stronger than being strongly cuspidal). Actually, it will be more convenient for us to work with functions that are not necessarily compactly supported: we will take f in the so-called Harish-Chandra-Schwartz space (see §1.3 for a reminder) denoted by CpGpF qq. The notions of strong cuspidality and of cusp forms extend verbatim to this bigger space. The following theorem, whose proof is scattered all over this paper (see Theorem 2.1.1, Theorem 3.1.1 and Theorem 4.1.1), is our main technical result:
Theorem 4 Let f P CpGpF qq be a strongly cuspidal function. Then, the expression defining Jχpf q is absolutely convergent (see Theorem 2.1.1) and we have:
(i) (see Theorem 4.1.1) A geometric expansion
Jχpf q “ ż
ΓellpHq
DHpxqΘfpxqχpxq´1dx
(ii) (see Theorem 3.1.1) If f is moreover a cusp form, a spectral expansion
Jχpf q “ ÿ
πPIrrsqrpGq
mpπ, χq Trace π_pf q
where IrrsqrpGq denotes the set of (equivalence classes of ) irreducible square-integrable
representations of GpF q and for all π P IrrsqrpGq, π_ is the smooth contragredient of π.
We prove this theorem by following closely the general method laid down by [46], [47] and [11]. In particular, a crucial point to get the spectral expansion in the above theorem is to show that for π square-integrable the abstract multiplicity mpπ, χq is also the multiplicity of π in the discrete spectrum of L2pHpF qzGpF qq. This fact is established in the course
of the proof of Proposition 3.2.1 using the simple adaptation of an idea that goes back to Sakellaridis-Venkatesh ([44] Theorem 6.4.1) and Waldspurger ([47] Proposition 5.6).
Here is an outline of the contents of the different parts of the paper. In the first part, we set up the main notations and conventions as well as collect different results that will be needed in the subsequent sections. It includes in particular a discussion of a natural generalization of Arthur’s pG, Mq-families to symmetric pairs that we call pG, M, θq-families. The second part contains the proof of the absolute convergence of Jχpf q for strongly cuspidal
functions f and in the third part we establish a spectral expansion of this distribution when f is a cusp form. These two parts are actually written in the more general setting of tempered symmetric pairs pG, Hq (which were called strongly discrete in [22]) to which the proofs extend verbatim. The fourth part deals with the geometric expansion of Jχpf q. There we
really have to restrict ourself to the setting of Galois pairs (that is when G “ RE{FHE) since
a certain equality of Weyl discriminants (see 4.1.1), which is only true in this particular case, plays a crucial role in allowing to control the uniform convergence of certain integrals. Finally, in the last part of this paper we prove the formula for the multiplicity (Theorem 3) and give two applications of it towards Prasad’s conjecture (Theorem 1 and Theorem 2).
1
Preliminaries
1.1
Groups, measures, notations
Throughout this paper we will let F be a p-adic field (i.e. a finite extension of Qp for a
certain prime number p) for which we will fix an algebraic closure F . We will denote by |.| the canonical absolute value on F as well as its unique extension to F . Unless specified otherwise, all groups and varieties that we consider in this paper will be tacitly assumed to be defined over F and we will identify them with their points over F . Moreover for every finite extension K of F and every algebraic variety X defined over K we will denote by RK{FX Weil’s restriction of scalars (so that in particular we have a canonical identification
pRK{FXqpF q “ XpKq).
Let G be a connected reductive group over F and AG be its maximal central split torus.
AG :“ X˚pAGq b R
where X˚pAGq denotes the abelian group of cocharacters of AG. If V is a real vector space
we will always denote by V˚ its dual. The space A˚
G can naturally be identified with
X˚pA
Gq b R “ X˚pGq b R where X˚pAGq and X˚pGq stand for the abelian groups of
algebraic characters of AG and G respectively. More generally, for every extension K{F
we will denote by X˚
KpGq the group of characters of G defined over K. There is a natural
morphism HG: GpF q Ñ AG characterized by
xχ, HGpgqy “ logp|χpgq|q
for all χ P X˚pGq. We set A
G,F :“ HGpAGpF qq. It is a lattice in AG. The same notations
will be used for the Levi subgroups of G (i.e. the Levi components of parabolic subgroups of G): if M is a Levi subgroup of G we define similarly AM, AM, HM and AM,F. We will
also use Arthur’s notations: PpMq, FpMq and LpMq will stand for the sets of parabolic subgroups with Levi component M, parabolic subgroups containing M and Levi subgroups containing M respectively. Let K be a maximal special compact subgroup of GpF q. Then, for all parabolic subgroups P with Levi decomposition P “ MU the Iwasawa decomposi-tion GpF q “ MpF qUpF qK allows to extend HM to a map HP : GpF q Ñ AM defined by
HPpmukq :“ HMpmq for all m P MpF q, u P UpF q and k P K. For all Levi subgroups M Ă L
there is a natural decomposition
AM “ ALM ‘ AL
where AL
M is generated by HMpKerpHL|M pF qqq and we will set aLM :“ dimpALMq. The Lie
algebra of G will be denoted by g and more generally for any algebraic group we will denote its Lie algebra by the corresponding Gothic letter. We will write Ad for the adjoint action of G on g. We denote by exp the exponential map which is an F -analytic map from an open neighborhood of 0 in gpF q to GpF q. For all subsets S Ă G, we write CentGpSq, CentGpF qpSq
and NormGpF qpSq for the centralizer of S in G, resp. the centralizer of S in GpF q, resp. the
normalizer of S in GpF q. If S “ txu we will denote by Gx the neutral connected component
of CentGpxq :“ CentGptxuq. We define Gregas the open subset of regular semisimple elements
of G and for all subgroups H of G we will write Hreg :“ H X Greg. Recall that a regular
element x P GregpF q is said to be elliptic if AGx “ AG. We will denote by GpF qell the set of
regular elliptic elements in GpF q. The Weyl discriminant DG is defined by
DGpxq :“
detp1 ´ Adpxq|g{gxq
For every subtorus T of G we will write
WpG, T q :“ NormGpF qpT q{ CentGpF qpT q
for its Weyl group. If A Ă G is a split subtorus we will denote by RpA, Gq the set of roots of A in G i.e. the set of nontrivial characters of A appearing in the action of A on g. More
generally, if H is a subgroup of G and A Ă H is a split subtorus we will denote by RpA, Hq the set of roots of A in H.
In all this paper we will assume that all the groups that we encounter have been equipped with Haar measures (left and right invariant as we will only consider measures on unimodular groups). In the particular case of tori we normalize these Haar measure by requiring that they give mass 1 to their maximal compact subgroups. For any Levi subgroup M of G we equip AM with the unique Haar measure such that volpAM{AM,Fq “ 1. If M Ă L are two
Levi subgroups then we give AL
M » AM{AL the quotient measure.
Finally, we will adopt the following slightly imprecise but convenient notations. If f and g are positive functions on a set X, we will write
fpxq ! gpxq for all x P X
and we will say that f is essentially bounded by g, if there exists a c ą 0 such that fpxq ď cgpxq, for all x P X
We will also say that f and g are equivalent and we will write fpxq „ gpxq for all x P X
if both f is essentially bounded by g and g is essentially bounded by f.
1.2
log-norms
All along this paper, we will assume that gpF q has been equipped with a (classical) norm |.|g, that is a map |.|g : gpF q Ñ R` satisfying |λX|g “ |λ|.|X|g, |X ` Y |g ď |X|g` |Y |g and
|X|g “ 0 if and only if X “ 0 for all λ P F and X, Y P gpF q. For any R ą 0, we will denote
by Bp0, Rq the closed ball of radius R centered at the origin in gpF q.
In this paper we will freely use the notion of log-norms on varieties over F . The concept of norm on varieties over local fields has been introduced by Kottwitz in [36] §18. A log-norm is essentially just the logarithm of a Kottwitz’s log-norm and we refer to [11] §1.2 for the basic properties of these log-norms. For convenience, we collect here the definition and basic properties of these objects.
First, an abstract log-norm on a set X is just a real-valued function x ÞÑ σpxq on X such that σpxq ě 1, for all x P X. For two abstract log-norms σ1 and σ2 on X, we will say that
σ2 dominates σ1 and we will write σ1 ! σ2 if
σ1pxq ! σ2pxq
for all x P X. We will say that σ1 and σ2 are equivalent if each of them dominates the other.
For an affine algebraic variety X over F , choosing a set of generators f1, . . . , fm of its F
σXpxq “ 1 ` log pmaxt1, |f1pxq|, . . . , |fmpxq|uq
for all x P X. The equivalence class of σX doesn’t depend on the particular choice of
f1, . . . , fm and by a log-norm on X we will mean any abstract log-norm in this equivalence
class. Note that if U is the principal Zariski open subset of X defined by the non-vanishing of Q P OpXq, then we have
σUpxq „ σXpxq ` log
`
2` |Qpxq|´1˘
for all x P U. More generally, for X any algebraic variety over F , choosing a finite covering pUiqiPI of X by open affine subsets and fixing log-norms σUi on each Ui, we can define an
abstract log-norm on X by setting
σXpxq “ inftσUipxq; i P I such that x P Uiu
Once again, the equivalence class of σX doesn’t depend on the various choices and by a
log-norm on X we will mean any abstract log-norm in this equivalence class.
We will assume that all varieties that we consider in this paper are equipped with log-norms and we will set σ :“ σG and σ :“ σG.
Let p : X Ñ y be a regular map between algebraic varieties then we have
σYpppxqq ! σXpxq
for all x P X. If p is a closed immersion or more generally if p is a finite morphism ([36] Proposition 18.1(1)) we have
σYpppxqq „ σXpxq
for all x P X. We say that p has the norm descent property (with respect to F ) if, denoting by pF the induced map on F -points, we have
σYpyq „ inf xPp´1F pyq
σXpxq
for all y P pFpXpF qq. By Proposition 18.3 of [36], if T is a subtorus of G then the projection
G ։ T zG has the norm descent property i.e. we have
σTzGpgq „ inf
tPT pF qσptgq
(1.2.1)
for all g P GpF q. In section 1.9 we will prove that for H an F -spherical subgroup of G (i.e. a subgroup such that there exists a minimal parabolic subgroup P0 of G with HP0 open) the
projection G ։ HzG also has the norm descent property.
Let T Ă G again be a maximal subtorus. As the regular map T zG ˆ Treg Ñ Greg,
σTzGpgq ! σpg´1tgq log
`
2` DGptq´1˘
(1.2.2)
for all g P G and all t P Treg.
For every variety X defined over F , equipped with a log-norm σX, and all M ą 0 we
will denote by Xră Ms, resp. Xrě Ms, the set of all x P XpF q such that σXpxq ă M, resp.
σXpxq ě M. With this notation, if T is a torus over F and k :“ dimpATq we have
volpT ră Msq ! Mk
(1.2.3)
for all M ą 0.
1.3
Function spaces
Let ω be a continuous character of AGpF q. We define SωpGpF qq :“ Cc8pAGpF qzGpF q, ωq
as the space of functions f : GpF q Ñ C which are smooth (i.e. locally constant), satisfy fpagq “ ωpaqf pgq for all pa, gq P AGpF qˆGpF q and are compactly supported modulo AGpF q.
Assume moreover that ω is unitary and let ΞG be Harish-Chandra Xi function associated
to a special maximal compact subgroup K of GpF q (see [45] §II.1). Then, we define the Harish-Chandra-Schwartz space CωpGpF qq as the space of functions f : GpF q Ñ C which
are biinvariant by an open subgroup of GpF q, satisfy fpagq “ ωpaqfpgq for all pa, gq P AGpF q ˆ GpF q and such that for all d ą 0 we have |f pgq| ! ΞGpgqσpgq´d for all g P GpF q.
1.4
Representations
In this paper all representations we will consider are smooth and we will always use the slight abuse of notation of identifying a representation π with the space on which it acts. We will denote by IrrpGq the set of equivalence classes of smooth irreducible representations of GpF q and by IrrcusppGq, IrrsqrpGq the subsets of supercuspidal and essentially
square-integrable representations respectively. If ω is a continuous unitary character of AGpF q we
will also write IrrωpGq (resp. Irrω,cusppGq, Irrω,sqrpGq) for the sets of all π P IrrpGq (resp.
π P IrrcusppGq, π P IrrsqrpGq) whose central character restricted to AGpF q equals ω. For all
π P IrrpGq we will denote by π_ its contragredient and by x., .y : π ˆ π_ Ñ C the canonical
pairing. For all π P Irrω,sqrpGq, dpπq will stand for the formal degree of π. Recall that it
depends on the Haar measure on GpF q and that it is uniquely characterized by the following identity (Schur orthogonality relations)
ż AGpF qzGpF q xπpgqv1, v1_yxv2, π_pgqv2_ydg “ 1 dpπqxv1, v _ 2yxv2, v_1y
for all v1, v2 P π and all v1_, v_2 P π_. From this, we easily infer that for every coefficient f of
Tracepπ_pf qq “ 1
dpπqfp1q (1.4.1)
Let π P IrrpGq and let ω be the inverse of the restriction of the central character of π to AGpF q. Then, for all f P SωpGpF qq we can define an operator πpf q on π by
xπpf qv, v_y :“ ż
AGpF qzGpF q
fpgqxπpgqv, v_ydg for all pv, v_q P π ˆ π_. For all f P S
ωpGpF qq this operator is of finite rank and a very deep
theorem of Harish-Chandra ([25] Theorem 16.3) asserts that the distribution
f P SωpGpF qq ÞÑ Tracepπpf qq
is representable by a locally integrable function which is locally constant on GregpF q. This
function, the Harish-Chandra character of π, will be denoted Θπ. It is characterized by
Tracepπpf qq “ ż
AGpF qzGpF q
Θπpgqf pgqdg
for all f P SωpGpF qq. If moreover the representation π is square-integrable (or more generally
tempered) with unitary central character, then the integral defining πpf q still makes sense for all f P CωpGpF qq, the resulting operator is again of finite rank and the above equality
continues to hold.
1.5
Weighted orbital integrals
Let M be a Levi subgroup and fix a maximal special compact subgroup K of GpF q. Using K we can define maps HP : GpF q Ñ AM for all P P PpMq (cf §1.1). Let g P GpF q. The
family
t´HPpgq; P P PpMqu
is a positive pG, Mq-orthogonal set in the sense of Arthur (see [2] §2). In particular, following loc. cit. using this family we can define a weight vMQpgq for all Q P F pMq. Concretely, vMQpgq is the volume of the convex hull of the set tHPpgq; P P PpMq, P Ă Qu (this convex hull
belongs to a certain affine subspace of AM with direction ALM where Q “ LU with M Ă L
and we define the volume with respect to the fixed Haar measure on AL
M). If Q “ G we set
vMpgq :“ vMGpgq for simplicity. For every character ω of AGpF q, every function f P CωpGpF qq
and all x P MpF q X GregpF q, we define, again following Arthur, a weighted orbital integral
by
ΦQMpx, f q :“ ż
GxpF qzGpF q
The integral is absolutely convergent by the following lemma which is an immediate conse-quence of 1.2.2 and Lemma 1.9.2 (which will be proved later).
Lemma 1.5.1 Let x P MpF q X GregpF q. Then, for all d ą 0 there exists d1 ą 0 such that
the integral ż
GxpF qzGpF q
ΞGpg´1xgqσpg´1xgq´d1σGxzGpxq
ddg
converges.
Once again if Q “ G, we will set ΦMpx, f q :“ ΦGMpx, f q for simplicity. If M “ G (so that
necessarily Q “ G), ΦGpx, f q reduces to the usual orbital integral.
1.6
Strongly cuspidal functions
Let ω be a continuous unitary character of AGpF q. Following [46], we say that a function
f P CωpGpF qq is strongly cuspidal if for every proper parabolic subgroup P “ MU of G we
have
ż
UpF q
fpmuqdu “ 0, @m P MpF q
(the integral is absolutely convergent by [45] Proposition II.4.5). By a standard change of variable, f is strongly cuspidal if and only if for every proper parabolic subgroup P “ MU and for all m P MpF q X GregpF q we have
ż
UpF q
fpu´1muqdu “ 0
We will denote by Cω,scusppGpF qq the subspace of strongly cuspidal functions in CωpGpF qq
and we will set Sω,scusppGpF qq :“ SωpGpF qq X Cω,scusppGpF qq. Let K be a maximal special
compact subgroup of GpF q. For x P GregpF q set Mpxq :“ CentGpAGxq (it is the smallest
Levi subgroup containing x). Then, by [46] Lemme 5.2, for all f P Cω,scusppGpF qq, all Levi
subgroups M, all Q P FpMq and all x P MpF q X GregpF q we have ΦQMpx, f q “ 0 unless
Q“ G and M “ Mpxq. For all x P GregpF q we set
Θfpxq :“ p´1q aG
MpxqΦ
Mpxqpx, f q
Then the function Θf is independent of the choice of K and invariant by conjugation ([46]
Lemme 5.2 and Lemme 5.3). Also by [46] Corollaire 5.9, the function pDGq1{2Θ
f is locally
bounded on GpF q.
We say that a function f P CωpGpF qq is a cusp form if it satisfies one of the following
equivalent conditions (see [45] Théorème VIII.4.2 and Lemme VIII.2.1 for the equivalence between these two conditions):
• For every proper parabolic subgroup P “ MU and all x P GpF q we have ż
UpF q
fpxuqdu “ 0;
• f is a sum of matrix coefficients of representations in Irrω,sqrpGq.
We will denote by 0C
ωpGpF qq the space of cusp forms. Let f P0CωpGpF qq and set fπpgq :“
Tracepπ_pg´1qπ_pf qq for all π P Irr
ω,sqrpGpF qq and all g P GpF q. Then, fπ belongs to 0C
ωpGpF qq for all π P Irrω,sqrpGpF qq ([24] Theorem 29) and we have an equality
f “ ÿ
πPIrrω,sqrpGq
dpπqfπ
(1.6.1)
(This is a special case of Harish-Chandra-Plancherel formula, see [45] Theorem VIII.4.2). Let 0S
ωpGpF qq :“ SωpGpF qq X 0CωpGpF qq be the space of compactly supported cusp
forms. Similar to the characterization of 0C
ωpGpF qq, a function f P SωpGpF qq belongs to 0S
ωpGpF qq if and only if it satisfies one the the following equivalent conditions:
• For every proper parabolic subgroup P “ MU and all x P GpF q we have ż
UpF q
fpxuqdu “ 0;
• f is a sum of matrix coefficients of representations in Irrω,cusppGq.
Moreover, for f P0S
ωpGpF qq, we have fπ P 0SωpGpF qq for all π P Irrω,cusppGq and a spectral
decomposition
f “ ÿ
πPIrrω,cusppGq
dpπqfπ
(1.6.2)
Finally, we will need the following proposition.
Proposition 1.6.1 Let π P IrrsqrpGq and let f be a matrix coefficient of π. Then, we have
Θfpxq “
1
dpπqfp1qΘπpxq for all xP GregpF q.
Proof: Unfortunately, the author has been unable to find a suitable reference for this probably well-known statement (however see [14] Proposition 5 for the case where x is elliptic and [2] for the case where π is supercuspidal). Let us say that it follows from a combination of Arthur’s noninvariant local trace formula ([5], Proposition 4.1) applied to the case where one of the test functions is our f and of Schur orthogonality relations. Note that Arthur’s local trace formula was initially only proved for compactly supported test functions, but see [6] Corollary 5.3 for the extension to Harish-Chandra Schwartz functions.
1.7
Tempered pairs
Let H be a unimodular algebraic subgroup of G (e.g. a reductive subgroup). We say that the pair pG, Hq is tempered if there exists d ą 0 such that the integral
ż
HpF q
ΞGphqσphq´ddh
is convergent. This notion already appeared in [22] under the name of strongly discrete pairs. Following the referee suggestion we have decided to call these pairs tempered instead so that it is more in accordance with the notion of strongly tempered pairs introduced by Sakellaridis-Venkatesh in [44] §6 (since the latter implies the former but not conversely). This terminology is also justified by the fact that pG, Hq is tempered if and only if the Haar measure on HpF q defines a tempered distribution on GpF q i.e. it extends to a continuous linear form on CpGpF qq. Moreover, by a result of Benoist and Kobayashi [8], when H is reductive and in the case where F “ R (which is not properly speaking included in this paper) a pair pG, Hq is tempered if and only if L2pHpF qzGpF qq is tempered as a unitary
representation of GpF q. Although the author has not checked all the details, the proof of Benoist and Kobayashi seems to extend without difficulties to the p-adic case. However, we propose here a quick proof of one of the implications (but we won’t use it in this paper). Proposition 1.7.1 Assume that the pair pG, Hq is tempered. Then, the unitary represen-tation of GpF q on L2pHpF qzGpF qq given by right translation is tempered i.e. the Plancherel
measure of L2pHpF qzGpF qq is supported on tempered representations.
Proof: We will use the following criterion for temperedness due to Cowling-Haagerup-Howe [17]:
(1.7.1) Let pΠ, Hq be a unitary representation of GpF q. Then pΠ, Hq is tempered if and only if there exists d ą 0 and a dense subspace V Ă H such that for all u, v P V we have
|pΠpgqu, vq| ! ΞGpgqσpgqd for all g P GpF q where p., .q denotes the scalar product on H. We will check that this criterion is satisfied for V “ C8
c pHpF qzGpF qq Ă H “ L2pHpF qzGpF qq.
Let ϕ1, ϕ2 P Cc8pHpF qzGpF qq and choose f1, f2 P Cc8pGpF qq such that
ϕipxq “
ż
HpF q
fiphxqdh
by g and by p., .q the L2-inner product on L2pHpF qzGpF qq, we have pRpgqϕ1, ϕ2q “ ż HpF qzGpF q ϕ1pxgqϕ2pxqdx “ ż HpF qzGpF q ż HpF qˆHpF q f1ph1xgqf2ph2xqdh2dh1dx “ ż HpF qzGpF q ż HpF qˆHpF q f1ph1xgqf2ph2h1xqdh2dh1dx “ ż GpF q ż HpF q f1pγgqf2phγqdhdγ
for all g P GpF q. Let d ą 0 that we will assume sufficiently large in what follows. As f1 and
f2 are compactly supported, there obviously exist C1 ą 0 and C2 ą 0 such that
|f1pγq| ď C1ΞGpγqσpγq´2d and |f2pγq| ď C2ΞGpγqσpγq´d
for all γ P GpF q. It follows that for all g P GpF q we have
|pRpgqϕ1, ϕ2q| ď C1C2 ż GpF q ż HpF q ΞGpγgqΞGphγqσphγq´ddhσpγgq´2ddγ
Since σpγ1γ2q´1 ! σpγ1q´1σpγ2q for all γ1, γ2 P GpF q, this last expression is essentially
bounded by σpgq2d ż GpF q ż HpF q ΞGpγgqΞGphγqσphq´ddhσpγq´ddγ
for all g P GpF q. Let K be the special maximal compact subgroup used to define ΞG. Then
ΞG is invariant both on the left and on the right by K and since σpk
1γk2q´1 ! σpγq´1 for all
γ P GpF q and k1, k2 P K, we see that the last integral above is essentially bounded by
σpgq2d ż GpF q ż HpF q ż KˆK ΞGpγk1gqΞGphk2γqdk1dk2σphq´ddhσpγq´ddγ
for all g P GpF q. By the ‘doubling principle’ ([45] Lemme II.1.3), it follows that
|pRpgqϕ1, ϕ2q| ! ΞGpgqσpgq2d ż GpF q ΞGpγq2σpγq´ddγˆ ż HpF q ΞGphqσphq´ddh
for all g P GpF q. By [45] Lemme II.1.5 and the assumption that pG, Hq is tempered, for d sufficiently large the two integrals above are absolutely convergent. Thus, the criterion of Cowling-Haagerup-Howe is indeed satisfied for V “ C8
c pHpF qzGpF qq and consequently
L2
pHpF qzGpF qq is tempered.
Finally, we include the following easy lemma which gives an alternative characterization of tempered pairs because it is how the tempered condition will be used in this paper.
Lemma 1.7.1 Set AH
G “ pAGX Hq0. The pair pG, Hq is tempered if and only if there exists
dą 0 such that the integral ż
AH
GpF qzHpF q
ΞGphqσphq´ddh
converges.
Proof: As ΞG is A
GpF q invariant, it clearly suffices to show:
(1.7.2) For d ą 0 sufficiently large, we have
σphq´3d ! ż AH GpF q σpahq´3dda! σphq´d for all h P HpF q.
For this, we need first to observe that
σphq „ inf
aPAH GpF q
σpahq (1.7.3)
for all h P HpF q. Indeed, as AH
GzH is a closed subgroup of AGzG, this is equivalent to the
fact that the projection H ։ AH
GzH has the norm descent property and this can be easily
deduce from the existence of an algebraic subgroup H1 of H such that the multiplication
morphism AH
G ˆ H1 Ñ H is surjective and finite (so that in particular H1pF q has a finite
number of orbits in AH
GpF qzHpF q).
By the inequalities σphq ! σpahq and σpaq ! σpahqσphq for all a P AH
GpF q and all
hP HpF q, for any d ą 0 we have ż AH GpF q σpahq´3dda! σphq´2d ż AH GpF q σpahq´dda! σphqdσphq´2dż AH GpF q σpaq´dda
for all h P HpF q. For d sufficiently large, the last integral above is absolutely convergent. Moreover, as the left hand side of the above inequality is clearly invariant by h ÞÑ ah for any aP AH
GpF q, by 1.7.3, for d sufficiently large we get
ż AH GpF q σpahq´3dda! ˆ inf aPAH GpF q σpahq ˙d σphq´2d! σphq´d
for all h P HpF q and this shows one half of 1.7.2. On the other hand, by the inequality σpahq ! σpaqσphq for all a P AH
GpF q and h P HpF q, for any d ą 0 we have
σphq´3d ż AH GpF q σpaq´3dda ! ż AH GpF q σpahq´3dda
for all h P HpF q. Once again, for d sufficiently large the two integrals above are absolutely convergent and, as the right hand side of the inequality is invariant by h ÞÑ ah for any aP AH
GpF q, by 1.7.3 for d sufficiently large we get
σphq´3d ! ˆ inf aPAH GpF q σpahq ˙´3d ! ż AH GpF q σpahq´3dda
for all h P HpF q and this proves the second half of 1.7.2.
1.8
Symmetric varieties
1.8.1 Basic definition, θ-split subgroups
Let H be an algebraic subgroup of G. Recall that H is said to be symmetric if there exists an involutive automorphism θ of G (defined over F ) such that
pGθq0 Ă H Ă Gθ
where Gθ denotes the subgroup of θ-fixed elements. If this is the case, we say that H and θ
are associated. The involution θ is not, in general, determined by H but by [26] Proposition 1.2, its restriction to the derived subgroup of G is. From now on and until the end of §1.8.2 we fix a symmetric subgroup H of G and we will denote by θ an associated involutive automorphism.
Let T Ă G be a subtorus. We say that T is θ-split if θptq “ t´1 for all t P T and we
say that it is pθ, F q-split if it is θ-split as well as split as a torus over F . For every F -split subtorus A Ă G we will denote by Aθ the maximal pθ, F q-split subtorus of A. A parabolic
subgroup P Ă G is said to be θ-split if θpP q is a parabolic subgroup opposite to P . If this is the case, HP is open, for the Zariski topology, in G (this is because h ` p “ g) and similarly HpF qP pF q is open, for the analytic topology, in GpF q. If P is a θ-split parabolic subgroup, we will say that the Levi component M :“ P X θpP q of P is a θ-split Levi subgroup. Note that this terminology can be slightly confusing since a torus can be a θ-split Levi without being θ-split as a torus (e.g. for G “ GLp2q, T the standard maximal torus and θ given by θpgq “ ˆ 1 1 ˙ g ˆ 1 1 ˙
). Nevertheless, the author believe that no confusion should arise in this paper as the context will clarify which notion is being used.
Actually, a Levi subgroup M Ă G is θ-split (i.e. it is the θ-split Levi component of a θ-split parabolic) if and only if M is the centralizer of a pθ, F q-split subtorus if and only if M is the centralizer of AM,θ. We will adapt Arthur’s notation to θ-split Levi and parabolic subgroups
as follows: if M is a θ-split Levi subgroup we will denote by PθpMq, resp. FθpMq, resp.
LθpMq the set of all θ-split parabolic subgroups with Levi component M, resp. containing
M, resp. the set of all θ-split Levi subgroups containing M. Let M Ă G be a θ-split Levi subgroup. We set
and aM,θ :“ dim AM,θ. Note that we have
AM “ AM,θ‘ AθM
where as before a θ superscript indicates the subset of θ-fixed points. This decomposition is compatible with the decompositions
AM “ ALM ‘ AL
for all L P LθpMq. Hence, we also have
AM,θ “ ALM,θ‘ AL,θ
for all L P LθpMq, where we have set AL
M,θ :“ AM,θ X ALM. Also, we let aLM,θ :“ dim ALM,θ “
aM,θ ´ aL,θ. We define an homomorphism HM,θ : MpF q Ñ AM,θ as the composition of
the homomorphism HM with the projection AM ։ AM,θ. For all P P PθpMq, the roots
RpAM,θ, UPq of AM,θ in the unipotent radical UP of P can be considered as elements of
the dual space A˚
M,θ of AM,θ. There is a unique subset ∆P,θ Ă RpAM,θ, UPq such that
every element of pAM,θ, UPq is in an unique way a nonnegative integral linear combination
of elements of ∆P,θ. The set ∆P,θ is the image of ∆P by the natural projection A˚M ։ A˚M,θ
and it forms a basis of pAG
M,θq˚. We call it the set of simple roots of AM,θ in P . Define
A`P,θ :“ tX P AM; xα, Xy ą 0 @α P ∆P,θu
Then, we have the decomposition
AM,θ “
ğ
QPFθpM q
A`Q,θ
More precisely the set RpAM,θ, Gq of roots of AM,θ in G divides AM,θinto certain facets which
are exactly the cones A`
Q,θ where Q P F
θpMq. In a similar way, the subspaces supporting
the facets of this decomposition are precisely the subspaces of the form AL,θ, L P LθpMq,
whereas the chambers (i.e. the open facets) are precisely the cones A`
P,θ for P P P θpMq.
Fixing a maximal special compact subgroup K of GpF q, for all P P PθpMq, we define a map
HP,θ : GpF q Ñ AM,θ
as the composition of HP with the projection AM ։ AM,θi.e. we have HP,θpmukq “ HM,θpmq
for all m P MpF q, u P UPpF q and k P K.
Let A0be a maximal pθ, F q-split subtorus and let M0 be its centralizer in G. For simplicity
we set A0 :“ AM0,θ. It is known that the set of roots RpA0, Gq of A0in G forms a root system
in the dual space `AG 0
˘˚
to AG
0 (Proposition 5.9 of [26]). The Weyl group associated to this
root system is naturally isomorphic to
and is called the little Weyl group (associated to A0) (again Proposition 5.9 of [26]). Two
maximal pθ, F q-split subtori are not necessarily HpF q-conjugate (e.g. for G “ GLn and
H “ Opnq) but they are always GpF q-conjugate 1.
Let M be a θ-split Levi subgroup and let α P RpAM,θ, Gq. Then we define a ‘coroot’
α_ P AG
M,θ as follows. First assume that α is a reduced root (i.e. α
2 R RpAM,θ, Gq). Let Mα
be the unique θ-split Levi containing M such that AMα,θ “ Kerpαq. Let Qα be the unique
θ-split parabolic subgroup of Mα with θ-split Levi M such that ∆Qα “ tαu. Let P
Mα
0 be a
minimal θ-split parabolic subgroup of Mα contained in Qα and set M0 :“ P0Mα X θpP Mα
0 q,
A0 :“ AM0,θ. Let ∆
Mα
0 be the set of simple roots of A0 in P0Mα. Then there is an unique
simple root β P ∆Mα
0 whose projection to A˚M,θ equals α. Let β_P A0 be the corresponding
coroot. Then we define α_as the image of β_by the projection A
0 ։ AM,θ. We easily check
that this construction does not depend on the choice of PMα
0 since for another choice P01 ,Mα with M1 0 :“ P01 ,Mα X θpP1 0 ,Mα q and A1 0 :“ AM1
0,θ there exists m P MpF q with mA
1 0m´1 “ A0 and mP1 0 ,Mα m´1 “ PMα
0 . If α is nonreduced, there exists α0 P RpM,θ, Gq such that α “ 2α0
and we simply set α_“ α_0
2 .
Let rα P RpAM, Gq be a root extending α and rα_P AM the corresponding coroot. Then, in
general the projection of rα_to A
M,θdoes not coincide with α_as defined above but, however,
the two are always positively proportional. Finally, we remark that when M is a minimal θ-split Levi subgroup, so that RpAM,θ, Gq is a root system, then for all α P RpAM,θ, Gq, α_
coincides with the usual coroot defined using this root system.
Let P be a θ-split parabolic subgroup. We set AP,θ :“ AM,θ and aP,θ “ aM,θ where
M :“ P X θpP q and we let ∆_
P,θ Ď A G
M,θ be the set of simple coroots corresponding to
∆P,θ Ď pAGM,θq˚ and p∆P,θ Ď pAGM,θq˚ be the basis dual to ∆_P,θ. More generally, let Q Ą P be
another θ-split Levi subgroup. We set AQ
P,θ :“ A L M,θ and a Q P,θ :“ a L M,θ where L :“ Q X θpQq and we let ∆Q P,θ Ď pA Q
P,θq˚ be the set of simple roots of AM,θ in P X L, p∆ Q
P,θq_ Ď A Q P,θ be
the corresponding set of simple coroots and p∆QP,θ Ď pAQP,θq˚ be the basis dual to p∆Q
P,θq_. We have decompositions AP,θ“ AQP,θ‘ AQ,θ, A˚P,θ “ pA Q P,θq ˚‘ A˚ Q,θ for which ∆Q P,θ Ď ∆P,θ, p∆ Q
P,θq_ Ď ∆_P,θ and moreover ∆Q,θ (resp. ∆_Q,θ) is the image of
∆P,θ´ ∆QP,θ (resp. ∆_P,θ´ p∆ Q
P,θq_) by the projection A˚P,θ ։ A˚Q,θ (resp. AP,θ ։ AQ,θ). We
define the following functions:
• τP,θQ : characteristic function of the set of X P AM,θsuch that xα, Xy ą 0 for all α P ∆QP,θ;
• pτP,θQ : characteristic function of the set of X P AM,θ such that x̟α, Xy ą 0 for all
̟α P p∆ Q P,θ;
1Indeed if A0and A1
0are two maximalpθ, F q-split tori, M0:“ CentGpA0q,M01 :“ CentGpA10q, P0P PθpM0q
and P1
0P PθpM01q then by [26] Proposition 4.9, P0and P01 are conjugated by an element of gP GpF q X HP0
• δQM,θ: characteristic function of the subset AL,θ of AM,θ.
We also define a function ΓQ
P,θ on AM,θ ˆ AM,θ, whose utility will be revealed in the next
section, by ΓQP,θpH, Xq :“ ÿ RPFθpM q;P ĎRĎQ p´1qaR,θ´aQ,θτR P,θpHqpτ Q R,θpH ´ Xq
Let M be a θ-split Levi subgroup. Then, for all P P PθpMq we set
A`,˚P,θ :“ tλ P A˚
P,θ; xα_, λy ą 0 @α_P ∆_P,θu
As for AM,θ, the set of coroots RpAM,θ, Gq_ divides A˚M,θ into facets which are exactly the
cones A`,˚
Q,θfor Q P FθpMq and the chambers for this decomposition are the A `,˚
P,θ, P P PθpMq.
As usual, we say that two parabolics P, P1 P PθpMq are adjacent if the intersection of the
closure of their corresponding chambers contains a facet of codimension one. If this is the case, the hyperplane generated by this intersection is called the wall separating the two chambers.
1.8.2 pG, M, θq-families and orthogonal sets
As we have recalled in the previous section, the combinatorics of θ-split Levi and parabolic subgroups is entirely governed, as is the case for classical Levi and parabolic subgroups, by a root system. As a consequence, for M a θ-split Levi subgroup of G the classical theory of pG, Mq-families due to Arthur extends without difficulty to a theory of pG, M, θq-families indexed by θ-split parabolics that we now introduce. By definition, a pG, M, θq-family is a family pϕP,θqPPPθpM q of C8 functions on iA˚M,θ such that for any two adjacent parabolic
subgroups P, P1 P PθpMq, the functions ϕ
P,θ and ϕP1,θ coincide on the wall separating the
chambers iA`,˚
P,θ and iA `,˚
P1,θ. To a pG, M, θq-family pϕP,θqPPPθpM q we can associate a scalar
ϕM,θ as follows: the function
ϕM,θpλq :“
ÿ
PPPθpM q
ϕP,θpλqεP,θpλq, λ P iA˚M,θ
where we have set
εP,θpλq :“ meas ` AG M,θ{Zr∆ _ P,θs ˘ ź α_P∆_ P,θ xλ, α_y´1 is C8 and we define ϕ
M,θ :“ ϕM,θp0q. Note that we need a Haar measure on AGM,θ for the
definition of the functions εP,θ (P P PθpMq) to make sense. We fix one as follows. Let AM,θ
be the image of AM,θ in G :“ G{AG and let A1M,θ be the inverse image of AM,θ in G. Let
AG
M,θ,F Ď A G
M,θ denote the image of A1M,θpF q by H G
M,θ. It is a lattice of A G
M,θ and we choose
our measure so that the quotient AG
We will actually only need pG, M, θq-families of a very particular shape obtained as follows. We say that a family of points YM,θ “ pYP,θqPPPθpM qin AM,θ is a pG, M, θq-orthogonal
set if for all adjacent P, P1 P PθpMq we have
YP,θ´ YP1,θ “ rP,P1α_
where rP,P1 P R and tα_u “ ∆_P,θX ´∆_P1,θ. We say that the family is a positive pG, M,
θq-orthogonal set if it is apG, M, θq-orthogonal set and moreover rP,P1 ě 0 for all adjacent P, P1 P
PθpMq. To a pG, M, θq-orthogonal set Y
M,θ “ pYP,θqPPPθpM qwe associate the pG, M, θq-family
pϕP,θp., YM,θqqPPPθpM q defined by
ϕP,θpλ, YM,θq :“ exλ,YP,θy, λP iA˚M,θ
and we let vM,θpYM,θq :“ ϕM,θp0, YM,θq be the scalar associated to this pG, M, θq-family. If
YM,θ is a positive pG, M, θq-orthogonal set then vM,θpYM,θq is just the volume of the convex
hull of the elements in the family YM,θ (with respect to the fixed Haar measure on AGM,θ).
For any pG, M, θq-orthogonal set YM,θ “ pYP,θqPPPθpM q and all Q P FθpMq we define YQ,θ to
be the projection of YP,θ to AQ,θ for any P P PθpMq with P Ă Q (the result is independent
of the choice of P ) and more generally for any Q, R P FθpMq with Q Ă R we let YR
Q,θ be the
projection of YP,θ to ARQ,θ for any P P P
θpMq with P Ă Q. Then, for any L P LθpMq the
family YL,θ :“ pYQ,θqQPPθpLq forms a pG, L, θq-orthogonal set.
Fixing a maximal special compact subgroup K of GpF q to define maps HP,θ(P P PθpMq),
for all g P GpF q the family YM,θpgq :“ p´HP,θpgqqPPPθpM q is a positive pG, M, θq-orthogonal
set. Indeed, the family p´HPpgqqPPPpM q is a positive pG, Mq-orthogonal set in the classical
sense of Arthur (see [2] §2) and thus for all P “ MUP, P1 “ MUP1 P PθpMq we have
´HPpgq ` HP1pgq P
ÿ
αPRpAM,UPqX´RpAM,UP 1q
R`α_.
As the projection of RpAM, UPq (resp. RpAM, UP1q) to AM,θ˚ is RpAM,θ, UPq (resp. RpAM,θ, UP1q)
and for all α P RpAM, Gq the projection of the coroot α_ to AM,θ˚ is positively proportional
to α_, where α P RpA
M,θ, Gq denotes the projection of α to A˚M,θ, it follows that
´HP,θpgq ` HP1,θpgq P
ÿ
αPRpAM,θ,UPqX´RpAM,θ,UP 1q
R`α_
for all P, P1 P PθpMq i.e. Y
M,θpgq is a positive pG, M, θq-orthogonal set.
We define
vM,θpgq :“ vM,θpYM,θpgqq
There is another easier way to obtain pG, M, θq-orthogonal sets. It is as follows. Let M0 Ă M
be a minimal θ-split Levi subgroup with little Weyl group W0. Fix P0 P PθpM0q. Then, for
all X P A0 :“ AM0,θ we define a pG, M0, θq-orthogonal set YrXs0 :“ pY rXsP01,θqP01PPθpM0q by
that wP1
0P0 “ P
1
0. By the general construction explained above this also yields a pG, M,
θq-orthogonal set YrXsM,θ “ pY rXsP,θqPPPθpM q.
Let YM,θ “ pYP,θqPPPθpM q be a pG, M, θq-orthogonal set. For Q P FθpMq we define a
function ΓQ M,θp., YM,θq on AM,θ by ΓQM,θpH, YM,θq :“ ÿ RPFθpM q;RĂQ δRM,θpHqΓQR,θpH, YR,θq
where the functions δR
M,θ and Γ Q
R,θ have been defined in the previous section. Let L “
QX θpQq. Fixing a norm |.| on AL
M,θ, we have the following basic property concerning the
support of this function (see [38] Corollaire 1.8.5):
(1.8.1) There exists c ą 0 independent of YM,θsuch that for all H P AM,θwith ΓQM,θpH, YM,θq ‰
0we have |HQ|ď c supPPPθpM q;P ĂQ|Y
Q
P,θ| where H
Qdenotes the projection of H to AL M,θ.
Moreover, if the pG, M, θq-orthogonal set YM,θ is positive then ΓQM,θp., YM,θq is just the
char-acteristic function of the set of H P AM,θ such that HQ belongs to the convex hull of
pYP,θQqPPPθpM q;P ĂQ([38] Proposition 1.8.7). Without assuming the positivity of our pG, M,
θq-orthogonal set, we have the identity ([38] Lemme 1.8.4(3))
ÿ
QPFθpM q
ΓQM,θpH, YM,θqτQ,θG pH ´ YQ,θq “ 1
(1.8.2)
for all H P AM,θ.
Let R be a free Z-module of finite type. Recall that a exponential-polynomial on R is a function on R of the following form
fpY q “ ÿ
χP pR
χpY qpχpY q
where pR denotes the group of complex (not necessarily unitary) characters of R and for all χP pR, pχ is a ‘complex polynomial’ function on R, i.e. an element of SymppC b Rq˚q,which
is zero for all but finitely many χ P pR. If f is an exponential-polynomial on R then a decomposition as above is unique, the set of characters χ P pR such that pχ‰ 0 is called the
set of exponents of f and p1(corresponding to χ “ 1 the trivial character) is called the purely
polynomial part of f . Finally, we define the degree of f as the maximum, over all χP pR, of the degree of the polynomials pχ. We record the following lemma whose proof is elementary:
Lemma 1.8.1 Let R be a free Z-module of finite type, let f be a exponential-polynomial on R and let C Ă R b R be an open cone (with a vertex possibly different from the origin). Then,if the limit
lim
YPRXC |Y |Ñ8
fpY q
exists it equals the constant term of the purely polynomial part of f .
Let M0 Ă M be a minimal θ-split Levi subgroup, A0 :“ AM0,θ and fix P0 P P
θpM 0q.
For all X P A0 we dispose of the pG, M, θq-orthogonal set YrXsM,θ “ pY rXsP,θqPPPθpM q
defined above. We let YM,θ` YrXsM,θ :“ pYP,θ ` Y rXsM,θqPPPθpM q be the sum of the two
pG, M, θq-orthogonal sets YM,θ and YrXsM,θ. Obviously, it is also a pG, M, θq-orthogonal set.
We let
rvM,θpYM,θ ` YrXsM,θq :“
ż
AGpF qzA1M,θpF q
ΓGM,θpHM,θpaq, YM,θ ` YrXsM,θqda
where we recall that A1
M,θ is the subtorus generated by AG and AM,θ. Let A0,F denote the
image of AM0,θpF q by HM0,θ. It is a lattice in A0 and we have the following lemma (combine
[41] Lemme 1.7(ii) with equalities 1.5(2) and 1.3(7) of loc.cit.):
Lemma 1.8.2 For every lattice RĂ A0,θ,FbQ the function X P R ÞÑ rvM,θpYM,θ`YrXsM,θq
is an exponential-polynomial whose degree and exponents belong to finite sets which are in-dependent of YM,θ. Moreover, if we denote by rvM,θ,0pYM,θ,Rq the constant coefficient of the
purely polynomial part of this exponential-polynomial there exists cą 0 depending only on R such that for all k ě 1 we have
rv M,θ,0pYM,θ, 1 kRq ´ vM,θpYM,θq ď ck´1 ˜ sup PPPθpM q |YP,θ| ¸aG M,θ
Let now YM “ pYPqPPPpM q be a usual pG, Mq-orthogonal set. This induces a pG, M,
θq-orthogonal set YM,θ :“ pYP,θqPPPθpM q where, for all P P PθpMq, we denote by YP,θ the
projection of YP to AM,θ. The subspace AM,θ` AG of AM being special in the sense of [3]§7 2, we have a descent formula (Proposition 7.1 of loc.cit.)
vM,θpYM,θq “ ÿ LPLpM q dGM,θpLqv Q MpYMq (1.8.3)
where for all L P LpMq, Q is a parabolic with Levi component L which depends on the choice of a generic point ξ P AM and dGM,θpLq is a coefficient which is nonzero only if
AG M “ A
G,θ
M ‘ ALM. Moreover if A G,θ
M “ 0 then we have dGM,θpGq “ 1. Let K be a special
maximal compact subgroup of GpF q that we use to define the maps HP for P P PpMq
2Indeed with the notations of loc.cit. we need to check that for every root β P RpAM,θ, Gq the sum
ř
αPΣpβqmαα is trivial on AθM but this is trivial since for all αP Σpβq we have ιpαq :“ ´θpαq P Σpβq and
and HP,θ for P P PθpMq. Then, the formula 1.8.3 applied to the particular case where
YP “ HPpgq for all P P PpMq and for some g P GpF q yields
vM,θpgq “ ÿ LPLpM q dGM,θpLqv Q Mpgq (1.8.4)
1.9
Estimates
In this section we collect some estimates that we will need in the core of the paper. We start with four lemmas concerning maximal tori and integrals over regular orbits in G.
Lemma 1.9.1 Let T Ă G be a maximal torus . Then, we have σptq ! σpg´1tgq
for all tP T and all g P G.
Proof: Let W :“ W pGF, TFq be the absolute Weyl group of T and set B :“ G{{G ´ Ad (i.e. the GIT quotient of G acting on itself by the adjoint action). Let p : G Ñ B be the natural projection. By Chevalley theorem, the inclusion T ãÑ G induces an isomorphism T {{W » B and thus the restriction of p to T is a finite morphism. Hence, we have
σptq „ σBppptqq
for all t P T and it follows that
σptq „ σBppptqq “ σBpppg´1tgqq ! σpg´1tgq
for all t P T and all g P G.
Lemma 1.9.2 (Harish-Chandra, Clozel) Let T Ă G be a maximal torus. Then, for all dą 0 there exists d1 ą 0 such that
DGptq1{2 ż
TpF qzGpF q
ΞGpg´1tgqσpg´1tgq´d1dg! σptq´d for all tP TregpF q.
Proof: By Corollary 2 of [14] there exists d0 ą 0 such that
sup tPTregpF q DGptq1{2 ż TpF qzGpF q ΞGpg´1tgqσpg´1tgq´d0dgă 8
Thus by Lemma 1.9.1, for all d ą 0 we have
DGptq1{2 ż
TpF qzGpF q
ΞGpg´1tgqσpg´1tgq´d0´ddg ! σptq´d
Lemma 1.9.3 Let T Ă G be a subtorus such that Treg :“ T X Greg is nonempty (i.e. T
contains nonsingular elements). Then, for all k ą 0, there exists d ą 0 such that the integral ż
TpF q
logp2 ` DGptq´1qkσptq´ddt converges.
Proof: We denote by X˚
FpT q the group of regular characters of T defined over F . There
exists a multiset Σ of nontrivial elements in X˚
FpT q such that
DGptq “ ź
αPΣ
|αptq ´ 1|
for all t P TregpF q where we have denoted by |.| the unique extension of the absolute value
over F to F . We have
logp2 ` DGptq´1q !ź
αPΣ
logp2 ` |αptq ´ 1|´1q Thus, by Cauchy-Schwartz, it suffices to prove the following claim:
(1.9.1) For all α P X˚
FpT q ´ t1u and all k ą 0 there exists d ą 0 such that the integral
ż
TpF q
log`2` |αptq ´ 1|´1˘kσptq´ddt converges.
Let α P X˚
FpT q ´ t1u and let Γα Ă ΓF be the stabilizer of α for the natural Galois action.
Write Γα “ GalpF {Fαq where Fα{F is a finite extension. By the universal property of
restriction of scalars, α induces a morphism rα : T Ñ RFα{FGm. Denoting by Kerprαq the
kernel of rα, for all k ą 0 and all d ą 0 we have ż TpF q log`2` |αptq ´ 1|´1˘kσptq´kdt“ ż TpF q{ KerprαqpF q log`2` |αptq ´ 1|´1˘k ż KerprαqpF q σptt1q´ddt1dt
As there exists a subtorus T1 Ă T such that the multiplication map T1 ˆ Kerprαq Ñ T is an
isogeny (so that aσptqσpt1q ! σptt1q and σptq „ σ
T{ Kerprαqptq for all pt, t1q P Kerprαq ˆ T1), we
see that for all d sufficiently large (i.e. so that the integral below converges) we have ż
KerprαqpF q
σptt1q´ddt1 ! σ
T{ Kerprαqptq´d{2
for all t P T pF q{ KerprαqpF q. Since T pF q{ KerprαqpF q is an open subset of pT { Kerprαqq pF q we are thus reduced to the case where rα is an embedding.
Define Npαq P X˚pT q by
Npαq :“ ź
σPΓF{Γα
σpαq
We distinguish two cases. First, if Npαq ‰ 1 then we have an inequality
log`2` |αptq ´ 1|´1˘! log`2` |Npαqptq ´ 1|´1˘
for all t P T pF q with Npαqptq ‰ 1. Hence, up to replacing α by Npαq we may assume that α P X˚pT q in which case by the previous reduction we are left to prove 1.9.1 in the
particular case where T “ Gm and α “ Id in which case it is easy to check. Assume now
that Npαq “ 1. Since rα is an embedding this implies that T is anisotropic and we just need to prove that for all k ą 0 the function
tP T pF q ÞÑ log`2` |αptq ´ 1|´1˘k
is locally integrable. Using the exponential map we are reduced to proving a similar statement for vector spaces where we replace αptq ´ 1 by a linear form which is easy to check directly.
Combining 1.2.2 with Lemma 1.9.2 and Lemma 1.9.3 we get the following:
Lemma 1.9.4 Let T Ă G be a subtorus such that Treg :“ T X Greg is nonempty and let TG
be the centralizer of T in G (a maximal torus). Then, for all k ą 0 there exists d ą 0 such that the integral
ż TpF q DGptq1{2 ż TGpF qzGpF q ΞGpg´1tgqσpg´1tgq´dσTGzGpgqkdgdt converges.
The following lemma will be needed in the proof of the next proposition. As it might be of independent interest we present it separately.
Lemma 1.9.5 Let G be an anisotropic group over F and Y an affine G-variety. Set Y1 “
Y{G for the GIT quotient (it is an affine algebraic variety over F ) and denote by p : Y Ñ Y1
the natural projection. Then we have
σYpyq „ σY1pppyqq
for all yP Y pF q.
Proof: First, we have σY1pppyqq ! σYpyq for all y P Y since p is a morphism of algebraic
varieties. Let f P F rY s, we need to show that
for all y P Y pF q. Let W be the G-submodule of F rY s generated by f and V be its dual. There is a natural morphism ϕ : Y Ñ V and we have a commutative diagram
Y ϕ //Y1 “ Y {G ϕ1 V //V1 :“ V {G
By definition there is a function fV P F rV s such that f “ fV ˝ ϕ and moreover σV1py1q !
σY1py1q for all y1 P Y1. Hence, we are reduced to the case where Y “ V and we may assume
that f is homogeneous. By Kempf’s extension of the stability criterion of Mumford over any perfect field ([30], Corollary 5.1) and since G is anisotropic, for every v P V pF q the G-orbit G.v Ă V is closed. It follows that there exist homogeneous polynomials P1, . . . , PN P
FrV1s “ F rV sG whose only common zero in V pF q is 0. We only need to show that for some
R, C ą 0 we have
maxp1, |f pvq|q ď C maxp1, |P1pvq|, . . . , |PNpvq|qR
(1.9.2)
for all v P V pF q. Up to replacing f, P1, . . . , PN by some powers, we may assume that they
are all of the same degree. Then, for every 1 ď i ď N, f{Pi is a rational function on the
projective space PpV q and the map
rvs P PpV qpF q ÞÑ min p|pf {P1qprvsq|, . . . , |pf {PNqprvsq|q P R`
is continuous for the analytic topology hence bounded (as PpV qpF q is compact) and this proves that inequality 1.9.2 is true for R “ 1 and some constant C.
Following [31] Definition 4.9, we say that a subgroup H Ă G is F -spherical if there exists a minimal parabolic subgroup P0 of G such that HP0 is open, in the Zariski topology, in G.
For example symmetric subgroups (see §1.8.1) are F -spherical. Recall that in §1.2 we have defined a ‘norm descent property’ for regular maps between F -varieties.
Proposition 1.9.1 Let H Ă G be an F -spherical subgroup. Then, the natural projection p : GÑ HzG has the norm descent property.
Proof: Set X :“ HzG. By [36] Proposition 18.2 (1), it suffices to show that X can be covered by Zariski open subsets over which the projection p has the norm descent property. Since Gacts transitively on X it even suffices to construct only one such open subset (because its G-translates will have the same property). By the local structure theorem ([31] Corollary 4.12), there exists a parabolic subgroup Q “ LU of G such that
• U “ HQ is open in G;
• H X Q “ H X L and this subgroup contains the non-anisotropic factors of the derived subgroup of L.
Obviously, to show that the restriction of p to U has the norm descent property it is sufficient to establish that L Ñ H X LzL has the norm descent property. We are thus reduced to the case where H contains all the non-anisotropic factors of the derived subgroup of G. Let Gder
denote the derived subgroup of G, Gder,ncthe product of the non-anisotropic factors of Gder,
Gder,c the product of the anisotropic factors of Gder and set G1 “ Gder,cZpGq0, H1 “ H X G1.
Then, we have H “ Gder,ncH1 and the multiplication map Gder,ncˆ G1 Ñ G is an isogeny. It
follows that there exists a finite set tγi; iP Iu of elements of GpF q such that
GpF q “ğ iPI Gder,ncpF qG1pF qγi and HpF qzGpF q “ď iPI H1pF qzG1pF qγi
From these decompositions, we infer that we only need to prove the norm descent property for G1 Ñ H1zG1 i.e. we may assume that G
der,nc “ 1. Let Gc be the product of Gder,c with
the maximal anisotropic subtorus of ZpGq0 and consider the projection
p1 : X :“ HzG Ñ X1 :“ HGczG
We claim that
σXpxq „ σX1pp1pxqq
(1.9.3)
for all x P XpF q. As H is reductive (since GpF q contains no unipotent element), X is affine and the claim follows from the Lemma 1.9.5.
Now because of 1.9.3, we may replace X by X1 i.e. we may assume that H contains G c. As
the multiplication map Gcˆ AG Ñ G is an isogeny, by a similar argument as before we are
reduced to the case where G is a split torus for which the proposition is easy to establish directly.
2
Definition of a distribution for all symmetric pairs
2.1
The statement
Let G be a connected reductive group over F , H be a symmetric subgroup of G and θ be the involution of G associated to H (see §1.8.1). Set AH
G “ pAG X Hq0, G :“ G{AG,
H :“ H{AH, X :“ AGpF qHpF qzGpF q, X :“ HAGzG, σX :“ σX and σ :“ σG. Note that
X is an open subset of XpF q. Let χ and ω be continuous unitary characters of HpF q and AGpF q respectively such that χ|AH
GpF q “ ω|A H
GpF q. Then, for all f P SωpGpF qq we define a
function Kχ
